As soon as Quantum Mechanics (QM) achieved a definite and coherent mathematical formulation, several physicists and philosophers claimed that its formal structure does not conform to the laws of classical propositional calculus.For an introduction and a clear discussion of quantum logic the reader may refer to cf. [1]}, [2]}, [3]}, [4]}. The first modifications of Classical Logic (CL) were advanced in the thirties in order to faithfully represent the physical content of quantum theory. For instance, in 1931 the Polish philosopher Zygmunt Zawirski proposed to apply Łukasiewicz's three-valued logic to QM, starting from considerations about the wave-particle duality and Heisenberg's uncertainty principle. In virtue of the latter, moreover, the logical empiricists Moritz Schlick and Philipp Frank claimed that the conjunction of two or more statements may have no meaning in quantum theory, concluding that the classical conjunction is not always valid in the context of quantum physics, as recalled in [5]}. In their view, a typical example of invalid conjunction is one attributing a definite value for both position and momentum to a single quantum particle at a given time \(t\) . Furthermore, in 1933 Fritz Zwicky suggested that QM rejects the law of excluded-middle.For historical details on quantum logic see [6]}. However, it is generally recognized that the founding text of Quantum Logic (QL) is Birkhoff and von Neumann's essay “The Logic of Quantum Mechanics” ([7]}), where the authors provided the standard account of the propositional calculus obeyed by quantum propositions.It is worth noting that von Neumann in his treatise Mathematische Grundlagen der Quantenmechanik published in 1932 anticipated that from the algebraic structure of quantum theory it would have been possible to formulate a new propositional calculus. These ideas would be fully expressed in his successive collaboration with Garrett Birkhoff.
